Let the three consecutive integers be x, x+1, and x+2. According to the given information, x + (x+1) + (x+2) = 39. Simplifying this equation, we get 3x + 3 = 39. Subtracting 3 from both sides, we get 3x = 36. Dividing both sides by 3, we get x = 12. Therefore, the largest integer among them is x+2 = 12+2 = 14.

To find the time at which the two trains meet, we need to calculate the relative speed between them. The relative speed is the speed at which the two trains are approaching each other. In this case, the relative speed is 70 km/h + 60 km/h = 130 km/h. Now, we can use the formula distance = speed x time to calculate the time taken for the two trains to meet. The distance between New Delhi and Mumbai is 1300 km, and the relative speed is 130 km/h. Therefore, the time taken for the two trains to meet is 1300 km / 130 km/h = 10 hours. Since the first train leaves New Delhi at 10:00 AM and the second train leaves Mumbai at 11:00 AM, the two trains will meet at 10:00 AM + 10 hours = 8:00 PM. However, since the question asks for the time in PM, the answer is 2:30 PM.

In the given code, each letter is shifted two positions forward in the alphabet. For example, 'A' is shifted to 'C', 'P' is shifted to 'R', and so on. Therefore, 'ORANGE' is written as 'PRSPHF' in the same code.

Profit = Selling Price - Cost Price. Profit Percentage = (Profit / Cost Price) x 100. Given that the profit percentage is 20% and the cost price is (\$100). Therefore, Profit = (20/100) x (\$100) = (\$20). Selling Price = Cost Price + Profit = (\$100) + (\$20) = (\$120).

Number of male employees = 60% of 100 = 60. Number of female employees = 100 - 60 = 40. Number of male employees who leave = 20% of 60 = 12. Number of female employees who leave = 30% of 40 = 12. Total number of employees who leave = 12 + 12 = 24. Percentage of total employees who leave = (24 / 100) x 100 = 24%.

Area of a rectangle = Length x Width. Therefore, the area of the room is 12 meters x 8 meters = 96 square meters.

Speed = Distance / Time. Distance covered by the train in passing the telegraph pole = Length of the train. Distance covered by the train in passing the platform = Length of the train + Length of the platform. Therefore, Speed of the train = (Length of the train + 200 meters) / 20 seconds. Since the length of the train is not given, we cannot calculate the exact speed of the train. However, we can say that the speed of the train is greater than 36 km/h (which is the speed required to cover 200 meters in 20 seconds) and less than 72 km/h (which is the speed required to cover 200 meters + the length of the train in 20 seconds).

Squaring both sides of (x - y = 3), we get ((x - y)^2 = 3^2). Expanding the square, we get (x^2 + y^2 - 2xy = 9). Substituting (x^2 + y^2 = 25) in this equation, we get (25 - 2xy = 9). Rearranging the equation, we get (2xy = 16). Dividing both sides by 2, we get (xy = 8). Now, adding (x + y) and (x - y), we get (2x = x + y + x - y = 10). Therefore, (x + y = 10).

Discount = Original Price - Discounted Price. Discount Percentage = (Discount / Original Price) x 100. Given that the discount percentage is 10% and the discounted price is (\$90). Therefore, Discount = (10/100) x (\$100) = (\$10). Original Price = Discounted Price + Discount = (\$90) + (\$10) = (\$100).

To find the time at which the two trains meet, we need to calculate the relative speed between them. The relative speed is the speed at which the two trains are approaching each other. In this case, the relative speed is 70 km/h + 60 km/h = 130 km/h. Now, we can use the formula distance = speed x time to calculate the time taken for the two trains to meet. The distance between Mumbai and Delhi is 1300 km, and the relative speed is 130 km/h. Therefore, the time taken for the two trains to meet is 1300 km / 130 km/h = 10 hours. Since the first train leaves Mumbai at 9:00 AM and the second train leaves Delhi at 10:00 AM, the two trains will meet at 9:00 AM + 10 hours = 7:00 PM. However, since the question asks for the time in PM, the answer is 2:30 PM.

In the given code, each letter is shifted two positions forward in the alphabet. For example, 'C' is shifted to 'E', 'A' is shifted to 'C', and 'T' is shifted to 'V'. Therefore, 'DOG' is written as 'FPI' in the same code.

Number of male employees = 60% of 120 = 72. Number of female employees = 120 - 72 = 48. Number of male employees who leave = 25% of 72 = 18. Number of female employees who leave = 35% of 48 = 16. Total number of employees who leave = 18 + 16 = 34. Percentage of total employees who leave = (34 / 120) x 100 = 30%.

Area of a rectangle = Length x Width. Therefore, the area of the room is 15 meters x 10 meters = 150 square meters.

Speed = Distance / Time. Distance covered by the train in passing the telegraph pole = Length of the train. Distance covered by the train in passing the platform = Length of the train + Length of the platform. Therefore, Speed of the train = (Length of the train + 250 meters) / 25 seconds. Since the length of the train is not given, we cannot calculate the exact speed of the train. However, we can say that the speed of the train is greater than 45 km/h (which is the speed required to cover 250 meters in 25 seconds) and less than 72 km/h (which is the speed required to cover 250 meters + the length of the train in 25 seconds).

Using the identity (a^3 - b^3 = (a - b)(a^2 + ab + b^2)), we can write (x^3 - y^3 = (x - y)(x^2 + xy + y^2)). Substituting (x - y = 3) in this equation, we get (27 = 3(x^2 + xy + y^2)). Dividing both sides by 3, we get (x^2 + xy + y^2 = 9). Now, squaring both sides of (x - y = 3), we get ((x - y)^2 = 3^2). Expanding the square, we get (x^2 + y^2 - 2xy = 9). Adding (x^2 + xy + y^2) and (x^2 + y^2 - 2xy), we get (2(x^2 + y^2) = 18). Therefore, (x^2 + y^2 = 9).